Well, the multiplicative inverse of $a$ is defined to be that value $b$ for which $a \times b = 1$, where $\times$ is the multiplication operation in the field/ring/group in question.
Because we're talking about the group of multiplication modulo 65537, that means that the problem is, given $a$, find $b$ such that $ab \bmod 65537 = 1$.
Now, the % operator ...

Such a weak key schedule was chosen since the key schedule "theory" was not well developed by that time. Designers just modified the key schedule of DES a bit. Remember that these key schedules had to be optimized for hardware, and any extra operation would cost something in terms of area.

You don't need to compute a multiplicative inverse to encrypt or decrypt, in IDEA. All you need is the ability to multiply modulo $2^{16}+1$. See How can I implement the "Multiplication Modulo" and "Addition Modulo" operations in IDEA?
Key generation involves computing a multiplicative inverse. One way to compute the multiplicative ...

The multiplication operation is indeed not uniquely reversible given just the output. But we also have one of the inputs, namely, the subkey. We can use that to reverse the multiplication.
Decryption for IDEA requires changing the subkeys in the key schedule. I didn't find a good description of IDEA online, so I went back to Applied Cryptography, 2nd ...

Below is a small ruby program which calculates the inverse with respect to the IDEA multiplication. The IDEA multiplication is defined on [0..65535] by identifying 0 with 65536 and multiplying mod 65537 (the 4-th Fermat prime). The IDEA-multiplication can be calculated with data-independent timing as you can see below in mult. The addition chain used by ...

Addition modulo $2^{16}$ just means, add the two numbers as you normally would, and subtract $2^{16}$ from the result until the sum is less than $2^{16}$. So suppose you wanted to add, say, 51995 and 29291 modulo $2^{16}$:
51995 + 29291 = 81286
Subtract 2^16 = 65536, you get 81286 - 65536 = 15750
This is less than 65536, so 51995 + 29291 = 15750 modulo ...